22nd International Symposium on Plasma Chemistry
July 5-10, 2015; Antwerp, Belgium
Application of intrinsic low dimensional manifold method for simplifying
plasma chemistry
T. Rehman1, K.S.C. Peerenboom1, W. Graef1, E.H. Kemaneci2 and J. van Dijk1
1
Department of Applied Physics, Eindhoven University of Technology, Eindhoven, the Netherlands
2
Ruhr University Bochum, Theoretical Electrical Engineering, Bochum, Germany
Abstract: Numerical simulation of plasma models involving large numbers of species and
reactions is computationally expensive. In addition, the amount of data generated is
massive, and the interpretation becomes difficult. One of the solutions to overcome these
problems is to employ Chemical Reduction Techniques (CRT) used in combustion
research. The CRT we study here is ILDM (Intrinsic Low Dimensional Manifold).
Keywords:
ILDM
plasma chemistry, numerical simulation, chemical reduction techniques,
1. Introduction
The complete numerical fluid description of a plasma
involves solving a coupled set of partial differential
equations. These equations are
spatio-temporal
continuity equations (mass, momentum and energy). A
plasma model containing large numbers of chemical
species and reactions yields a high chemical complexity
and computational load. An example is a plasma model
for conversion of methane into higher hydrocarbons
consisting of 36 species and 367 gas phase reactions [1].
Due to the large number of species and reactions the
numerical
simulation
of
a
model
becomes
computationally expensive. In addition to high
computational load the amount of data generated is
massive, and the interpretation becomes difficult. Plasma
physics is not the only branch of science that deals with
these problems. Another discipline that encounters a
similar problem is combustion research. An example
from combustion research is the detailed computation of a
1-D laminar flame [2]. To overcome the difficulty
associated with high chemical complexity and expensive
computation, the combustion community employs
Chemical Reduction Techniques. Some examples of such
techniques are ILDM (Intrinsic Low Dimensional
Manifold), TGLDM (Trajectory Generated Low
Dimensional Manifold), and PCA (Principle Component
Analysis).
In this study we investigate the ILDM technique. The
ILDM technique uses the fact that a system of chemical
reactions can be studied by taking a fewer number of slow
reactions, from a complete set of reactions occurring in a
system. Other reactions are so fast that the variation in
system due to the fast reactions occurs very quickly.
Hence the fast reactions will quickly attain the steadystate and the full system can be analyzed by the slow
reactions. Based on this information the ILDM method
identifies a lower dimensional space (manifold) inside the
complete composition space. After a short interval of
time the fast time-scale processes will quickly move onto
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this lower dimensional manifold and the slow time-scale
processes will move the system tangential to the manifold
or along the manifold (as shown in Fig. 1) to finally reach
the equilibrium point.
Fig.1. Evolution of trajectories (T1 to T7) from different
initial conditions in the state space for the Zel'dovich
system [3]. The temperature is 1600 K. All the trajectories
quickly move towards a 1D line (shown in black) and then
along the line to the equilibrium point (labelled as Eq. in
the figure). The 1D line on which the trajectories bundle
together is the one-dimensional manifold. The system is
described by a 5D composition space (as there are five
species). Shown here is the projection of 5D space on a
2D plane of N and NO.
Identification of a Low Dimensional Manifold allows
decoupling of slow and fast time scales. A full system
1
description can be given by this lower dimensional
manifold without any significant loss in chemical kinetics.
The advantage of ILDM over conventional reduced
mechanisms (like the Partial Equilibrium assumption and
Quasi Steady State Assumption) is that it extracts the
required information automatically from a full system
description. The only input that has to be given is the
dimension of the manifold and the set of user defined
parameters. It does not rely on the so called intuition or
experience of a modeller to distinguish the fast and slow
reactions.
By constructing a low dimensional manifold the
reaction space is described in terms of only a few
parameters and it becomes possible to tabulate the results
in terms of those few parameters. Once the look-up table
is generated, the continuity equations are solved only for a
few parameters. The remaining values are read directly
from the table.
In this work we apply the method of ILDM (as
explained in section 2) to the simple case of the
Zel'dovich system from combustion engineering. The
ILDM method is then used for the reduced simulation of
an argon plasma. The reduced simulation from ILDM is
compared to the calculation which is done using CRM
(Collisional Radiative Model) [5] (as explained in section
4). The results are also compared with PCA (Principle
Component Analysis).
2. Mathematical Model for ILDM
The mathematical model for a system [2] consists of a
set of N (N = 2+n s ) equations namely, the particle
balances of the species (n s ) and the two equations that
define temperature and pressure. The particle balance is
written as
�⃗
∂y
= �S⃗(ψ
�⃗),
∂t
where �ψ⃗ = (T, P, �y⃗) with T the temperature, P the
pressure, �y⃗ the set of densities for n s species, and �S⃗ is the
source term.
For each point in the complete state space the
eigenvalues of the Jacobian of the source term are given
as
∂Si
.
Jij =
∂ψj
The eigenvalues and eigenvectors are obtained from
diagonalization of the Jacobian matrix. The eigenvalues λ
of the Jacobian matrix define n s characteristic time scales
associated with each species. The time scale t is given as
1
ti = .
λi
The eigenvectors define the characteristic directions
associated with the particular time scale. There are three
possibilities for how a system reacts to a small
perturbation. If the real part of the eigen value is
i. positive: perturbation will increase,
ii. 0: perturbation will not change,
iii. negative: perturbation will relax to 0.
2
If the eigenvalues are imaginary there will be
oscillation around the equilibrium point. The real part of
the eigenvalues will decide whether the oscillation
increases, decreases, or remains constant. In a physical
system all the eigenvalues are 0 and/or negative, yet
imaginary and non-zero eigenvalues might appear in
iterative approaches.
Repeated eigenvalues in a Jacobian matrix can cause
linearly dependent eigenvectors. In most cases
diagonalization is not possible because of the presence of
linearly dependent eigenvectors. Due to the difficulty
associated with the diagonalization, it is convenient to use
another basis namely, a basis of Schur vectors denoted as
� [4]. By Schur vector decomposition of the Jacobian we
Q
get
� T J� �Q = M
�,
Q
� is an upper triangular matrix with eigenvalues
where M
on the diagonal arranged in descending order of
� T is the transpose of the Schur vector
magnitude and Q
matrix.
Fig. 2. Figure shows the behaviour of the source vector
(shown in red) and the fast Schur vector (shown in blue)
in the vicinity of the 1D manifold (shown in green). The
source vectors turn and follow along the manifold and the
fast Schur vectors are perpendicular to the manifold.
The system will quickly reach the manifold curve due
to the fast processes and the slow time-scale processes
will move the system tangential to the manifold. In other
words the slow Schur vectors are parallel to the manifold.
Since these vectors are also orthogonal to each other, the
fast Schur vectors will be perpendicular to the manifold.
The source vector will move tangential to the manifold.
The source vector and the fast Schur vectors will become
perpendicular on the manifold. (as shown in fig.2)
The equation for a manifold is thus defined in terms of
inner product of the fast Schur vectors and the source,
� TL . �S⃗(ψ
�⃗) = 0,
Q
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� TL is the matrix containing fast Schur vectors. Q
� TL
where Q
is formed by neglecting the slow Schur vectors, that is
� T from the top (given in more detail in
n c +n p rows in Q
[4]). n c refers to the number of conserved quantities, and
n p refers to the number of user defined parameters. From
the equation of a manifold it can be seen that additional
2+n c +n p equations are needed to solve the system. The
remaining equations are the element conservation
equations, the parameter equations, and two additional
equations for temperature and pressure. The remaining
equations are given by
i. temperature
T − Tref = 0,
R
R
P
ii. pressure
P − � ni k B T = 0,
iii. elemental conservation equation
f�χi � − τi = 0,
where f�χi � is the function for calculating the total
number of element χ in the system from all the species
present in the system. Since an element can neither be
created nor destroyed in a system, the total number of
atoms of a particular element is fixed. This fixed value for
an element is τ.
iv. parameter equation
φi − pi = 0,
where φi refers to the parameter and pi is the value of the
parameter. A choice can be made for the number of
parameters and which species or which quantity should be
taken as parameter. The number of parameters define the
dimension of the manifold inside the complete statespace.
All the equations are solved simultaneously to get the
manifold point. By incrementing and/or decrementing the
values of parameter/parameters and solving the above set
of equations, we get a series of points that lie on the
manifold. It is convenient to start from the equilibrium
point because the equilibrium point will definitely be on
the manifold curve.
3. Application of ILDM to Zel'dovich system
The ILDM method, as explained in section 2, is
applied to the Zeldovich system [3] containing five
species (namely N, NO, N 2 , O and O 2 ) and two reversible
reactions namely
N + NO ⇆ N 2 + O,
and
N + O 2 ⇆ NO + O.
The temperature and the pressure of the system are
constant. The temperature is set at 1600 K and the
pressure is set to 73.16 MPa. Since temperature and
pressure are fixed, the total number of moles is fixed
(from the ideal gas law), so we can write
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� ni − � ni eq = 0,
where ni eq is the total moles at equilibrium.
There are two elements in the system N and O. So the
equations for the conservation of elements will be
CN + CNO + 2CN2 − τN = 0,
and
CNO + CO + 2CO2 − τO = 0,
where Cs is the concentration of species s and τ is the
fixed value for the particular element. Hence there are
three equations for conservation.
For a 1D manifold one of the parameters is chosen out
of the five species. In this case N :
φN − pN = 0,
where 𝑝𝑁 is the value that is changed for each new point
on the manifold. Since it is a five species system, five
equations are needed to fully solve the system. The
system of equations is completed by adding the manifold
equation. Points on the manifold are calculated by
changing the value of the parameter. A 1D manifold is
constructed as shown in Fig 1.
The manifold is described by a lookup table as
discrete points. Once the lookup table is generated the
continuity equations are solved only for fewer species, the
values for remaining species are directly read from the
lookup table. The reduced simulation for the Zel'dovich
system is compared to the full system calculation.
For reduced simulation only one parameter was calculated
and the rest of the values were read from the look-up
table. The comparison of the reduced simulation with the
full simulation is shown in Fig. 3.
Fig.3. Comparison of the concentration of N and NO as
calculated from the full simulation (shown in red) and the
reduced simulation (shown in green) for the Zel'dovich
system.
From Fig. 3 we can see that if the initial period of time
(time interval in the range of nanoseconds) is neglected,
the complete description of the system can be given by
generating a lookup table and doing the reduced
simulation.
3
4. Collisional Radiative Model
In a CRM the density of each level is determined by
the processes of collision and radiation [6]. In a CRM
described by Graef [5], the different excited levels of the
system can be categorized into Local Chemistry (LC) and
Transport Sensitive (TS) levels. For most of the excited
species the chemical processes will be much faster than
transport. These levels are categorized as LC levels. Other
levels typically the ion and the ground level have a much
higher density compared to the LC levels and fall in the
category of TS levels. By applying the Quasi Steady State
approximation we can distinguish the LC levels from the
TS levels and write LC levels in terms of TS levels. In
order to apply the Quasi Steady State approximation in a
CRM, the modeller must make a conscious choice for
each level in the model as to whether it is TS or LC level.
In contrast the ILDM method is a technique that
automatically separates TS levels from LC levels. The TS
levels and the LC levels found by ILDM are compared to
CRM, as well as the reduced simulation from both the
methods are compared.
[2] U. Maas and S.B. Pope, Combustion and Flame, 88,
239-264 (1992).
[3] Nicholas J. Glassmaker, Intrinsic Low-Dimensional
Manifold Method for Rational Simplification of Chemical
Kinetics-Undergraduate research (1999).
[4] Rafael E. Petrosyan, Developments of the Intrinsic
Low Dimensional Manifold method and Application of
the method to a Model of the Glucose Regulatory SystemUndergraduate research (2003).
[5] Wouter Graef, Zero-Dimensional Models For Plasma
Chemistry (2012).
[6] J.J.A.M. van der Mullen, Excitation Equilibria in
Plasmas; A Classification (1989).
5. Application of ILDM to plasma
The Zel'dovich system (as described in section 3) is a
problem from combustion engineering to study the
production of the pollutant NO. A plasma has additional
difficulties like the presence of charged species (ions and
electrons), electromagnetic field etc. Unlike combustion
systems, plasma in general may not be in thermal
equilibrium. Electrons will be at a higher temperature
than heavy particles. The presence of electromagnetic
field adds more parameters such as, the ambipolar electric
field and the magnetic field. Since, a plasma is electrically
neutral, the quasi-neutrality imposes an additional
constraint other than the elemental constraint. Our aim is
to apply the ILDM technique to the plasma, overcoming
these problems.
ILDM is applied to an argon plasma model. The argon
plasma we study contains excited states of an argon atom.
The processes that are taken into account are electron
excitation, heavy particle excitation and radiative
transitions. The comparison is made between CRM and
ILDM as well as PCA for an argon model.
6. Conclusion
The approach of ILDM can be used to simplify the
complex plasma chemistry by reducing the dimension of
the state space. The total number of species which define
the system are also reduced since the composition space is
defined in terms of only a few parameter. Hence the
description of full system can be studied in terms of few
parameters thereby reducing the computational load and
time.
7. References
[1] Christophe De Bie, Bert Verheyde, Tom Martens, Jan
van Dijk, Sabine Paulussen and Annemie Bogaerts,
Plasma Processes and Polymers, 8, 1033-1058 (2011).
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